Determine if the given set is a subspace of ℙ3. Justify your answer:The set of all polynomials of the form p(t)=at^3, where a is in ℝ.
To determine if a given set is a subspace of ℙ3, we need to check if it satisfies three conditions: closure under addition, closure under scalar multiplication, and contains the zero vector
To determine if a given set is a subspace of ℙ3, we need to check if it satisfies three conditions: closure under addition, closure under scalar multiplication, and contains the zero vector.
1. Closure under addition:
Let’s take two polynomials, p₁(t) = a₁t³ and p₂(t) = a₂t³, where a₁, a₂ ∈ ℝ.
The sum of these two polynomials is p(t) = p₁(t) + p₂(t) = a₁t³ + a₂t³ = (a₁ + a₂)t³.
This is still in the form of p(t) = at³, where a = a₁ + a₂, which is an element of ℝ.
Therefore, the set is closed under addition.
2. Closure under scalar multiplication:
Let p(t) = a₁t³ be a polynomial in the set, where a₁ ∈ ℝ.
If we multiply p(t) by a scalar c ∈ ℝ, we get cp(t) = c(a₁t³) = (ca₁)t³.
This is still in the form of p(t) = at³, where a = ca₁, which is an element of ℝ.
Thus, the set is closed under scalar multiplication.
3. Contains the zero vector:
The zero vector in ℙ3 is the polynomial p(t) = 0.
By letting a = 0, we have p(t) = at³ = 0t³ = 0, which is a polynomial of the desired form.
Therefore, the set contains the zero vector.
Since the given set satisfies all three conditions – closure under addition, closure under scalar multiplication, and contains the zero vector – it is indeed a subspace of ℙ3.
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